Point Slope Form for the Equation for a Line

Slope

We should already be familiar with the definition of slope in many of it's forms. Here are some of them:

$Slope=m=\frac{rise}{run}=\frac{change\:in\:y}{change\:in\:x}=\frac{\Delta y}{\Delta x}=\frac{y_2-y_1}{x_2-x_1}$

If we assume that we know that a point has a slope $m,\:$and goes through a point $\left(x_1,y_1\right),\:$we can use the definition of slope above to say that for any point $\left(x,y\right)$ on this line:

$m=\frac{y-y_1}{x-x_1}$

Multiplying both sides of this equation by $\left(x-x_1\right)\:$gives us:

$m\left(x-x_1\right)=y-y_1$

Which we can rewrite as;

Point Slope Form for the Equation for a Line

$y-y_1=m\left(x-x_1\right)\:$is an equation for a line with slope $m\:$that passes through the point$\left(x_1,y_1\right).$

Example

Write the equation for a line in point slope form that passes through (2,3) and has a slope of 7.

Solution

In this case, $\left(x_1,y_1\right)=\left(2,3\right)\:$and $m=7.$ Substituting into $y-y_1=m\left(x-x_1\right)\:$gives $y-3=7\left(x-2\right).$

Example

Write an equation for a line in slope-intercept form that goes through (1,-1) and (5,4).

Solution

In this case, we are not given the slope, but we can use the definition of the slope and the fact that the line goes through (1,-1) and (5,4) to find the slope.

$m=\frac{y_2-y_1}{x_2-x_1}=\frac{4-\left(-1\right)}{5-1}=\frac{4+1}{4}=\frac{5}{4}$

Now we can use the point slope form for a line with $m=\frac{5}{4}\:$and through $\left(1,-1\right).$

$y-\left(-1\right)=\frac{5}{4}\left(x-1\right)$

$y+1=\frac{5}{4}x-\frac{5}{4}$

Subtracting 1 from both sides gives:

$y=\frac{5}{4}x-\frac{5}{4}-1=\frac{5}{4}x-\frac{5}{4}-\frac{4}{4}=\frac{5}{4}x-\frac{9}{4}$

So, the equation for the line in slope-intercept form would be:

$y=\frac{5}{4}x-\frac{9}{4}$